models the homotopy colimit of the diagram \(|F| \colon {{C}\mathrm {art}}^\mathrm {op}\rightarrow {\Delta {T}\mathrm {op}}\) , for any \(F \in {H}_\infty \) .
Therefore, the functors they present on the \(\infty \) -categories underlying \({H}_\infty ^p\)  and \({\Delta {T}\mathrm {op}}\)  are equivalent, and they are further equivalent to the \(\infty \) -colimit functor.
Consequently, on the level of the underlying \(\infty \) -categories they each present left-adjoints to the functor that sends a space to the constant presheaf whose value is that space.
This means that both \(Re\)  and \(S_e\)  provide explicit presentations for the left-adjoint \(\Pi \)  in the three-fold adjunction which implements the cohesive structure on \({\mathbf {H}}\)  (see [1]}).
This appears to have been known for \(Re\) , but the functor \(S_e\)  has not been formally identified as a model for the cohesion functor \(\Pi \)  (although this has been indicated in [2]}).
